OK Tflfi STRUCTURE OP CRYSTALS. 307 



The assemblage of Bravaia ia therefore clearly identical with the 

 parallelepipedal network of points already referred to, which had been 

 investigated by Frankenheim. • 



2. The modifications of these types of symmetry which are introduced 

 by employing, in place of the points, symmetrical figures {polyedres) 

 possessing a symmetry less than that of the parallelepipedal network,^ 

 though compatible with it — e.g., by forming a cubic network of tetrahedral 

 particles similarly and appropriately oi'ientated. 



Thus in following Bravais' arguments with regard to assemblages we 

 note that, as a rule, he ignores for the moment any modifying or destruc- 

 tive effect exerted by the shape of the units {polyedres) on the elements 

 of symmetry."* He first treats a system as consisting only of the centres 

 of the units, and after the elements of symmetry of the system thus re- 

 garded have been established, he considers the effect of the shajie of the 

 units ; ^ this comes out in his definition of ' faces de meme espece.' He 

 says : ' We will distinguish by the term, faces of the same kind, as we 

 have done in the theory of assemblages, those which can be brought into 

 coincidence, row on row, by a suitable rotation or translation, the coin- 

 cidence of the faces including with it that of the assemblages. If, more- 

 over, the coincidence includes also that of the molecular polyhedra 

 which may be supposed to lie on the planes of those faces and to par- 

 ticipate in their movements, we may say that the faces are of the same 

 kind, and, moreover, identical.' '' The bodies employed as units have 

 in every case uniform orientation and one which is as symmetrical as 

 possible. 



As to the number of kinds of symmetrical arrangement possible 

 included under the first head, he says : ' The degree of symmetry of 

 an assemblage is characterised by the number of the axes of symmetry 

 which it possesses, the order of the symmetry of these axes and their 

 relative situation.' ^ As stated above, he distinguishes fourteen forms, 

 and assigns these to seven classes or systems, according to the number 

 and nature of the axes of symmetry which pass through a given node 

 (noeud) or point of the space-lattice.'' The anorthic space-lattice of 

 fig. 4 possesses only centro-symmetry ; if its angles were all right 

 angles it would possess the symmetry of the ortho-rhombic system ; 

 if, in addition, its edges were equal it would be a cubic lattice. The 

 similar bodies are called by Bravais in his later work polyhedra 

 (polyedres) ; in his earlier work on point-systems he speaks of them as 

 summits {sommeis), and suggests that for convenience of thought they be 

 regarded as having some small dimensions. Their size and shape are, 

 however, in this work generally kept in abeyance, although, before 

 concluding, he refers to the important effects of their shape or composite 

 structure in producing hemihedral and other partial forms. ^ Indeed, 

 according to Bravais' view, the symmetry of the assemblage is actually 

 determined by that of the molecule or unit.^ 



' Etudes CrlstaUograjiIdques, p. 103. ' Ibid., p. 194. ' Hid., p. 103. 



* This method has been pushed to its extreme by WulfiE and Blasius. Comp. 

 Schonflies, Krystallsystcme u. Krystallstructur, p. 320. 



' Etudes Cristallugraphiqves, p. lOG. " Ibid., p. 104. 



' Compt. Rend., 1849, xxix p. 135. 



» Ibid., 1848, xxvii. p. 603. Oomp. Journ. dc I'^cole Pohjteohniiiue, 1850, six. p. 

 127 ; and Etudes Crist ullograpMqucs, p. 103. 



' £titdcs Cristallugraplnques, p. 202. 



X2 



